The local cohomology of the Jacobian ring

Abstract

We study the 0-th local cohomology module $H_{\mathbf{m}}^0(R(f))$ of the jacobian ring $R(f)$ of a singular reduced complex projective hypersurface $X$, by relating it to the sheaf of logarithmic vector fields along $X$. We investigate the analogies between $H_{\mathbf{m}}^0(R(f))$ and the well known properties of the jacobian ring of a nonsingular hypersurface. In particular we study self-duality, Hodge theoretic and Torelli type questions for $H_{\mathbf{m}}^0(R(f))$.